Is there a tool to export/import ALL modifications I made to Precious Lite Pro template (sections, write-ups, etc..)? I tried using the included Export (and then Import) tool from WordPress however it does not export and import the design modifications; It only export pages and posts.

Or.. Which directory and files should I copy to clone the entire web site?

Yes you can import the pages and posts but you cannot import theme sections content and other data but there is an another solutions is to export the sql file of the database so you can get all theme options data in sql file.

There is no need to copy the theme files for taking data but yes you can copy the files if you made some changes on html and css.